On the Sources of the Height–Intelligence Correlation: New Insights from a Bivariate ACE model with Assortative Mating
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Citation Beauchamp, Jonathan P., David Cesarini, Magnus Johannesson, Erik Lindqvist, and Coren Apicella. 2010. On the sources of the height-intelligence correlation: New insights from a bivariate ACE model with assortative mating. Behavior Genetics 41(2): 242-252.
Published Version doi://10.1007/s10519-010-9376-7
Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:5141365
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ORIGINAL RESEARCH
On the sources of the height–intelligence correlation: New insights from a bivariate ACE model with assortative mating
Jonathan P. Beauchamp • David Cesarini • Magnus Johannesson • Erik Lindqvist • Coren Apicella
Received: 30 December 2009 / Accepted: 14 June 2010 / Published online: 6 July 2010 Ó The Author(s) 2010. This article is published with open access at Springerlink.com
Abstract A robust positive correlation between height and For both traits, we find suggestive evidence of a shared genetic intelligence, as measured by IQ tests, has been established in architecture with height, but we demonstrate that point esti- the literature. This paper makes several contributions toward mates are very sensitive to assumed degrees of assortative establishing the causes of this association. First, we extend the mating. Third, we report a significant within-family correla- standard bivariate ACE model to account for assortative tion between height and intelligence ðq^ ¼ 0:10Þ; suggesting mating. The more general theoretical framework provides that pleiotropy might be at play. several key insights, including formulas to decompose a cross- trait genetic correlation into components attributable to Keywords Assortative mating Bivariate ACE model assortative mating and pleiotropy and to decompose a cross- Genetic correlation Height Intelligence IQ trait within-family correlation. Second, we use a large dataset Within-family correlation of male twins drawn from Swedish conscription records and examine how well genetic and environmental factors explain the association between (i) height and intelligence and (ii) Introduction height and military aptitude, a professional psychogologist’s assessment of a conscript’s ability to deal with wartime stress. A robust positive correlation between height and intelli- gence, as measured by IQ tests,1 has been established in the literature with little consensus regarding its cause (Humph- Edited by Stacey Cherny. reys et al. 1985; Johnson 1991; Kanazawa and Reyniers 2009; Tanner 1979; Teasdale et al. 1989; Wheeler et al. J. P. Beauchamp (&) 2004). Both environmental and genetic explanations have Department of Economics, Harvard University, Cambridge, been advanced. For instance, previous research has found MA 02138, USA e-mail: [email protected] that markers of prenatal quality and nutritional status during childhood are associated with height (Eide et al. 2005; D. Cesarini Steckel 1995) and cognition in adulthood (See Gomez- Department of Economics, New York University, New York, Pinilla 2008; Martyn et al. 1996; Seidman et al. 1992; NY 10012, USA Sørensen et al. 1997), suggesting that early environmental M. Johannesson factors may be responsible for the correlation between height Department of Economics, Stockholm School of Economics, and intelligence (Abbott et al. 1998). Other evidence in Box 6501, Stockholm 113 83, Sweden support of environmental channels is the decline in the cor- E. Lindqvist relation between height and intelligence over time observed Research Institute of Industrial Economics, P.O. Box 55665, in the Scandinavian countries (Teasdale et al. 1989; Sundet Stockholm 102 15, Sweden et al. 2005; Tuvemo et al. 1999). On the other hand, reported
C. Apicella Department of Health Care Policy, Harvard Medical School, 1 We follow the literature and use the term ‘‘intelligence’’ to refer to Boston, MA 02115, USA what those IQ tests measure. 123 Behav Genet (2011) 41:242–252 243 heritability estimates in adults for measures of intelligence hypothesis that pleiotropy (or linkage) accounts for part of (e.g. 75%) (Neisser et al. 1996) and height (e.g. 87–93%) the genetic correlation. (Silventoinen et al. 2003) are high, suggesting that the relationship may instead be genetically mediated. For example, growth hormone deficiency, which is sometimes Method caused by genetic mutations, is characterized by both short stature and cognitive impairments (van Dam et al. 2005). Sample Evidence of a shared genetic architecture between brain volume, intracranial space, and height (Posthuma et al. Our data comes from two main sources: the Swedish Twin 2000) may also be interpreted as evidence of genetic medi- Registry (STR) and the Swedish National Service Adminis- ation of the height–intelligence correlation. tration (SNSA). The STR contains information on nearly all Two studies, using twin data, have attempted to estimate twin births in Sweden since 1886, and has been described in the components of the height–intelligence correlation that are further detail elsewhere (Lichtenstein et al. 2006). The due to genetic and environmental effects. The results were sample includes those individuals who have participated in at mixed. Silventoinen et al. (2006) found in several samples of least one of the Twin Registry’s surveys. The primary data- Dutch twins that the association between height and intelli- source is SALT (Screening Across the Lifespan Twin study). gence is primarily genetic in origin. Sundet et al. (2005), using This was a survey administered to all Swedish twins born conscription data from a considerably larger and more rep- between 1926 and 1958 and attained a response rate of 74%. resentative sample of Norwegian twins, found that the asso- Fifty percent of the subjects in the dataset are from the SALT ciation between height and intelligence is primarily explained cohort. The secondary source is the web-based survey by common environmental factors. Both papers use a standard STAGE (The Study of Twin Adults: Genes and Environ- bivariate ACE model to arrive at these conclusions. ment). This was a web-based survey administered between The present study makes several contributions toward November 2005 and March 2006 to all twins born in Sweden elucidating the sources of covariation between height and between 1959 and 1985. It attained a response rate of 61%. intelligence. First, we extend the bivariate ACE model to Approximately 30% of our subjects are drawn from STAGE. allow for assortative mating and derive a general formula for Our final datasource comes from a survey sent out in 1973 to decomposing a cross-trait genetic correlation into compo- the same cohort as SALT (Lichtenstein et al. 2006). nents attributable to assortative mating and pleiotropy. This We matched the Swedish twins to the conscription data provides a clear theoretical framework for studying the provided by SNSA. All Swedish men are required by law to correlation between two traits and is particularly useful when participate in a nationwide military conscription at the age of examining phenotypes for which there is high assortative 18. Before 1990, exemptions were very rare. The actual mating, height and intelligence being prime examples. The drafting procedure can take several days during which model demonstrates how sensitive estimates from the recruits undergo medical and psychological examination. bivariate ACE model are to assumptions about assortative The basic structure of the administered intelligence test has mating. Second, we use a sample of male Swedish twins remained unchanged during our study period, though minor matched to conscription records to examine the relative changes took place in 1980 and 1994. Recruits take four importance of genetic and environmental factors in subtests (logical, verbal, spatial and technical) which, for explaining the association between height and intelligence. most of the study period, are graded on a scale from 0 to 40. The sample used is the largest to date for such a study. It These raw scores are converted to a ordinal variable ranging includes an additional measure of cognitive function other from 1 to 9. Carlstedt (2000) discusses the history of psy- than intelligence, measured during enlistment through chometric testing in the Swedish military and provides interviews by a military psychologist. This measure, which evidence that this test of intelligence is a good measure of we label military aptitude, has a strong predictive validity for general intelligence. Thus, this test differs from the AFQT, labor market outcomes independent of intelligence, such as which focuses more on ‘‘crystallized’’ intelligence. wages, earnings and unemployment (Lindqvist and Vestman All conscripts also see a psychologist for a structured forthcoming). We apply the standard bivariate ACE model to interview. The psychologist has access to background infor- decompose the height–intelligence and the height–military mation on the interviewee, such as school grades, medical aptitude correlations, but we caution that the resulting esti- background, cognitive ability and answers to a battery of mates are very sensitive to assumptions about assortative questions on friends, family, and life. In conducting the mating and illustrate the sensitivity of these estimates by interview, the psychologist is required to follow a manual and, reporting results for different assumed levels of assortative ultimately, to make an assessment of the prospective recruit’s mating. Lastly, we report a significant within-family height– capacity to handle stress in a war situation. In making the intelligence correlation ðÞq^ ¼ 0.10 ; consistent with the assessment, the psychologist considers an individual’s ability 123 244 Behav Genet (2011) 41:242–252 to function in a group, adapt to new environments, as well as Table 1 Background variables his persistance and emotional stability. Motivation for doing MZ DZ Population the military service is not among the set of characteristics that is considered beneficial for succeeding in the military. The Income (in SEK) 342,631 335,987 325,245 psychologist assigns each interviewee an ordinal score from 1 SD 241,060 340,271 258,867 to 9, but again these are constructed from four raw scores, this Education (years) 12.50 12.14 12.54 time ranging from 1 to 5. Like the intelligence test score, the SD 2.68 2.63 2.35 military aptitude score is subject to measurement error 1 if married 0.50 0.51 0.51 because of random influences on conscript performance and SD––– because conscripts may differ in their motivation for the Age in 2005 48.85 51.69 45.43 2 military service. Lindqvist and Vestman (forthcoming) pro- SD 7.66 6.37 3.63 vide a more detailed description of the personality measure Intelligence 5.12 4.94 5.13 used by SNSA. SD 1.88 1.94 1.94 For both intelligence and military aptitude, the raw scores Military aptitude 5.32 5.17 5.08 underlying an individual’s ordinal score are available to us. SD 1.69 1.78 1.78 The raw scores are percentile rank-transformed and then Height (in cm) 178.45 178.60 178.94 converted by taking the inverse of the standard normal dis- SD 6.60 6.46 6.56 tribution to produce normally distributed test scores. The Note: Income (fo¨rva¨rvsinkomst) is defined as the sum of income transformation is done separately for each year, but when less earned from wage labor, income from own business, pension income than 100 pairs are available for a particular year, two adjoint and unemployment compensation. Capital income is not included. years are pooled. Since the sample of women who enlist in The education variable produced by Statistics Sweden is categorical the military comprises a small and highly self-selected group, (with seven categories ranging from middle school to PhD). The categorical scores are converted to years of education using the this paper focuses exclusively on men. We restrict the sample imputation model of Isacsson (2004). Population data was estimated to all male twin pairs for whom complete data on cognitive using information from Statistics Sweden on a representative sample ability and military aptitude is available for both twins. This born between 1954 and 1965. In our analyses, we do not use the leaves 1246 pairs of monozygotic (MZ) twins and 1568 pairs ordinal scores but normalized scores computed from subscores for intelligence and military aptitude, as described in the text of dizygotic (DZ) twins for analysis, all of which are born between 1950 and 1976. Descriptive statistics for the twins in our sample are presented in Table 1,disaggregatedby Yi ¼ aAi þ cCi þ eEi; zygosity. For expositional convenience, we report the ordinal where a, c, and e are 2 9 2 matrices of coefficients and instead of the normalized scores in the Table, even though where Ai Ai ; Ai 0; Ci Ci ; Ci 0 and Ei Ei ; Ei 0 the latter are used in the analyses that follow. ¼ 1 2 ¼ 1 2 ¼ 1 2 are, respectively, the latent additive genetic, common environmental, and individual environmental factors The bivariate ACE model, assortative mating underlying traits Yi and Yi : Throughout, we make the and the cross-trait genetic correlation 1 2 standard assumption that A, C, and E are mutually The bivariate ACE model independent. The model assumes that all genetic variance is additive, thereby ruling out dominance and epistasis. Suppose further, without loss of generality, that all the We follow biometrical genetic theory (Falconer and Mac- variables have been standardized to have mean zero and kay 1981) and previous papers on this subject (Silventoinen unit variance. The correlation between the traits of two et al. 2006; Sundet et al. 2005) to decompose the variance individuals i and j will then be equal to of a phenotype into shares attributable to additive genetic factors, common environment, and individual environment. E YiYj0 ¼ aE AiAj0 a0 þ cE CiCj0 c0 þ eE EiEj0 e0; ð1Þ Our empirical analysis uses the standard bivariate ACE i i i 0 model (Neale and Cardon 1992). Let Y ¼ Y1; Y2 be a where E denotes the expectations operator. It follows from vector of two observable phenotypes of individual i, and the standardization that the expectation of the product of suppose that two variables is simply their correlation. Table 2 summa- 0 0 rizes a set of assumptions about E[AiAj ], E[CiCj ] and i j0 2 However, the military aptitude score is subject to an additional form E[E E ]. These assumptions are customary in the behavior of measurement error since psychologists will vary in their judgement genetics literature, except for the presence of the CM matrix of identical conscripts. Lilieblad and Sta˚hlberg (1977) estimated the which, as we will show, accounts for assortative mating at correlation between the SNSA psychologists’ assessment to be .85 after letting thirty SNSA psychologists listen to tape recordings of the additive genetic level. The usual bivariate ACE model thirty enlistment interviews. does not account for assortative mating and thus implicitly 123 Behav Genet (2011) 41:242–252 245
Table 2 Assumed values of E[AiAj0], E[CiCj0] and E[EiEj0] for dif- identified. The substantive implication of the diagonality ferent relationships between individuals I and J assumption is that, while the latent factors underlying the Relationship between individuals i and j two traits may be correlated, each latent factor may only influence its respective trait.3 The restriction that C ¼ 0 i = j MZ twins DZ twins Unrelated M means that there is no assortative mating (including cross- i j0 1 4 E[A A ] CA CA 2 ðÞCA þ CM 0 trait assortative mating) for phenotypes Y1 and Y2. Below, i j0 E[C C ] CC CC CC 0 we also discuss the consequence of assuming different, i j0 E[E E ] CE 00 0more realistic values of CM:
i j0 Estimates of the shares of the observed variance in Note: This table shows the assumptions made about E[A A ], 2 E[CiCj0], E[EiEj0] within individuals, between MZ twins, between either trait attributable to additive genetic factors (a ), DZ twins and between unrelated individuals common environment (c2), and unique environment (e2) can be obtained from the above parameters. Examination assumes that C is a zero matrix. In the next section of the of Eq. 1 and some algebra reveals that M hi paper, we show that assortative mating enters the model i 2 2 2 2 VarðYkÞ¼E Yk ¼ 1 ¼ðakkÞ þðckkÞ þðekkÞ : this way, through CM: Behavior geneticists have previously studied models 2 2 2 2 2 2 We thus see that ak ¼ akk; ck ¼ ckk; and ek ¼ ekk ðk ¼ which allow for assortative mating, but they were mostly 1; 2Þ: It also follows from Eq. 1 that the within-individual concerned with the case of a single phenotype (see, for cross-trait correlation is given by instance, Eaves et al. 1978; Eaves and Heath 1981; Martin corr Y ; Y E Yi Yi et al. 1986; Keller et al. 2009). Eaves et al. (1984) consider ð 1 2Þ¼ 1 2 a a q c c q e e q : 2 the multivariate case and develop a model in which there is ¼ 11 22 A þ 11 22 C þ 11 22 E ð Þ assortative mating on a latent phenotype. We add to this The shares of the cross-trait phenotypic correlation attrib- literature by deriving formulas that describe, among others, utable to additive genetic factors, common environment, and the effects of assortative mating on the genetic correlation unique environment are thus a11a22qA ; c11c22qC ; and corrðY1;Y2Þ corrðY1;Y2Þ between two traits and by augmenting the bivariate ACE e11e22qE ; respectively.5 These shares depend on the corre- corrðY1;Y2Þ model to account for these effects. lations between the latent factors underlying both traits (qA, C The matrices in Table 2 have the following elements: qC,andqE) and on the shares of observed phenotypic var- 2 2 2 1 qA 1 qC 1 qE iance explained by each of the latent factors (a , c ,ande ). CA ¼ ; CC ¼ ; CE ¼ qA 1 qC 1 qE 1 Assortative mating in the bivariate ACE model m11 m 12 and CM ¼ : m 12 m22 We now show that assortative mating enters the bivariate ACE model as described in the previous section. For this, it As we show in the next section, m ¼ m12þm21 ; where m is 12 2 kl will be useful to augment the model to include individual the correlation between the latent additive genetic factors i’s father and mother. We assume that we are in genetic underlying phenotype Yk in fathers and phenotype Yl in mothers (k,l {1, 2}). Thus, there is positive assortative 3 An alternative way to proceed is to assume that the matrices a, c, mating at the additive genetic level for phenotypes k and l and e are lower triangular (i.e. that a12 = c12 = e12 = 0) and that if mkl [ 0 and/or mlk [ 0. qA = qC = qE = 0, while maintaining the assumption that CM ¼ 0: The resulting model has 19 free parameters: qA, qC, The substantive implication of the first assumption is that the latent q , m , m , m , m and 4 free parameters for each factors A1, C1, and E1 of the first trait Y1 can affect the second trait E 11 12 21 22 Y , whereas the latent factors A , C , and E of the second trait Y of the matrices a, c and e. However, only nine moments 2 2 2 2 2 cannot affect the first trait Y1. The second assumption implies that the can be computed from the data: the cross-trait covariance latent factors of the two traits are not correlated, even within between the MZ twins and, for each trait, the covariance individuals. The model resulting from this alternative set of assump- between the MZ twins; the corresponding three moments tions is sometimes referred to as a Cholesky decomposition. That model spans the same space as our preferred model, and it is thus for the DZ twins; and the population variances of the two possible to transform the parameters from either model into the traits as well as the population cross-trait covariance. Since parameters of the other (see Loehlin 1996, for a more thorough the number of parameters exceeds the number of inde- discussion). pendently informative equations, at least ten identifying 4 Alternatively, that restriction implies that if there is assortative assumptions need to be made. In the standard decomposi- mating, it does not have consequences at the genetic level. This might be the case, for example, if social homogamy fully explained the tion, it is assumed that a, c, and e are diagonal (i.e. that spousal phenotypic resemblance. a12 = a21 = c12 = c21 = e12 = e21 = 0) and that CM ¼ 0: 5 The term ‘‘share’’ in this context can be misleading, as correlations Under these assumptions, the remaining parameters are can be negative and these ‘‘shares’’ can therefore be negative. 123 246 Behav Genet (2011) 41:242–252 equilibrium and that all the parameters of the model are genetic assortative mating correlations CM are not directly fixed across generations. We can write observable, so it would be useful to have a mapping 1 relating CM to observable parameters. Unfortunately, to our E Ai jAFi ; AMi ¼ AFi þ AMi k k k 2 k k knowledge, no one has yet derived such a mapping for the ð3Þ general case of unconstrained multivariate assortative 1 F M ¼) Ai ¼ A i þ A i þ i ; 9 10 k 2 k k k mating. Gianola (1982) considers two special cases of i interest to the livestock industry. The first case is when where, as above, Ak is the latent additive genetic factor i assortative mating is actively practiced on one phenotype underlying phenotype Yk ðk ¼ 1; 2Þ for individual i; Fi and i only and a second phenotype is genetically correlated with Mi refer to i’s father and mother, respectively; and k is an the first due to pleiotropy (or linkage)—such as when, for error term independent of the parental genotypes. We instance, large bulls are mated with large cows, and the rewrite the error term as second trait is genetically correlated with cattle size. The i ¼ h Si ; ð4Þ second case is when mating pairs are assorted to have a k k k hi certain correlation between phenotype X in males and where h is a normalizing constant and E Si 2 1; thus k k ¼ phenotype Y in females—such as when, to use Gianola’s adhering to the convention of working with standardized example, high milk production females are mated to fast variables. The expression h Si is the deviation due to k k growing males. Assortative mating for height and intelli- Mendelian segregation (Otto et al. 1994). Finally, we let gence in humans is more complex and unlikely to fit either E Si Si q .6 The first equality in Eq. 3 implies that 1 2 ¼ S of these cases. We leave the derivation of a general map- E Si AFi E Si AMi E Si 0; and thus E Si AFi kj k ¼ kj k ¼ k ¼ k k ¼ ping to future research. However, the above discussion E Si AMi 0: For any pair of full siblings (including DZ k k ¼ should make it clear that it is important to investigate how twins) i and j, it follows that " sensitive the results from the standard bivariate decompo- 1 sition are to the assumption that C is equal to zero for E Ai A j ¼ E AFi þ AMi þ h Si M k l 2 k k k k traits with high assortative mating. # We now turn our attention to three additional features of 1 Fj Mj j the model which merit further exploration. First, we show Al þ Al þ hlSl 2 that in the augmented model (in which the elements of the hihi CM matrix are not constrained to be zero), the correlation 1 Fi Fj Mi Mj ¼ E Ak Al þ E Ak Al between two traits can be decomposed into parts attribut- 4 hihi able to assortative mating, pleiotropy, common environ- Fi Mj Mi Fj þ E Ak Al þ E Ak Al ment, and individual environment. Second, we investigate the bias which arises if a standard bivariate ACE model is 1 1 ¼ q þ q þ m þ m ¼ q þ m ; estimated in the presence assortative mating. Finally, we 4 A;kl A;kl kl lk 2 A;kl kl consider how within-family correlations can be used to i j shed light on the sources of a phenotypic correlation. where the second equality holds because Sk and Sl are uncorrelated with each other and with the other variables.7 Decomposition of qA The third equality follows from the fact that Fi = Fj and Mi = Mj and from ourhi assumption of genetic equilibrium, 2 which implies that E Ai ¼ 1 is constant across gen- Observe that k "# i i i i hi 2 erations and thus that E AkAl ¼ Cov Ak; Al ¼ qA;kl: 2 1 1 ¼ E Ai ¼ E AFi þ AMi þ h S Here, qA,kl is the correlation between the latent additive k 2 k k k k genetic factors underlying phenotypes k and l.8 Therefore, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i j0 1 E½ ¼A A ðÞC þ C : 1 2 1 mkk 2 A M ¼ ðÞþ1 þ m ðÞh ¼) h ¼ ; ðk ¼ 1; 2Þ; So far, we have only considered assortative mating at 2 kk k k 2 the additive genetic level. The parameters of the matrix of and that 6 i i i 0 In a more realistic model, the distribution of S ¼ S1; S2 would vary as a function of AFi and AMi : For simplicity and tractability, we 9 pObserveffiffiffiffiffiffiffiffiffiffiffiffi that it is not correct to simply assume that mkl ¼ do not consider such a model. 2 2 akk all rkl; where rkl is the phenotypic spousal correlation for traits 7 i j Sk and Sk are uncorrelated with each other because, conditional on k and l (k, l {1, 2}). To see, consider Gianola’s first case below. the parents’ genomes, the precise genetic draw from one DZ twin There, m 12 [ r12 because the phenotypic cross-trait correlation does not affect that of the other. follows from the genetic cross-trait correlation, and not the other 8 way around. Thus, qA,kl = 1ifk = l and qA,kl = qA if k = 1 and l = 2 as in the 10 previous section. Also, we use m kl to denote mkl when k = l. We are thankful to Peter Visscher for directing us to this paper. 123 Behav Genet (2011) 41:242–252 247 i i A lower bound for the share of the genetic correlation qA ¼ E A1A2 1 1 ¼ E AFi þ AMi þ h S AFi þ AMi þ h S ; Observe that for a pair of MZ twins, 2 1 1 1 1 2 2 2 2 2 hi MZ 0 0 0 E YiYj ¼ aCAa þ cCCc ; 1 ¼ ðÞþq þ m h h q : 2 A 12 1 2 S and that for a pair of DZ twins, hi 1 It follows that EDZ Y Y0 ¼ aðÞC þ C a0 þ cC c0; i j 2 A M C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ m þ ð1 m Þð1 m Þq : ð5Þ implying that A 12 11 22 S 0 MZ i j0 DZ i j0 0 Therefore, in equilibrium, the correlation between the aCAa ¼ 2 E Y Y E Y Y þ aCMa : latent additive genetic factors underlying phenotypes Y 1 Computing the off-diagonal elements of the symmetric and Y is equal to the sum of a term accounting for cross- 2 matrices on both sides of the above equation gives trait assortative mating and a term accounting for the MZ i j DZ i j genetic correlation arising from pleiotropy (or linkage). In a11a22qA ¼ 2 E Y1Y2 E Y1Y2 þ a11a22m 12: the limiting case where there is no assortative mating, Interestingly, when there is no cross-trait assortative q ¼ q : Without assortative mating, the Ajm11¼m 12¼m22¼0 S mating at the genetic level ðÞm ¼ 0 ; the share of the genetic correlation must be entirely attributable to 12 cross-trait phenotypic correlation attributable to additive pleiotropy (or linkage), and thus qs is the genetic genetic factors a11a22qA does not depend on the same- correlation that is attributable to pleiotropy (or linkage). corrðY1;Y2Þ trait assortative mating genetic correlations (m and m ). We can thus rewrite Eq. 5 as 11 22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Thus, under the maintained assumptions of the model, if cross-trait assortative mating is nonnegative at the genetic qA ¼ m 12 þ ð1 m11Þð1 m22ÞqA;Pleiotropy ð6Þ level ðÞm 12 0 ; estimates from the standard bivariate ACE In the limiting case where there is no pleiotropy, model still provide a lower bound for the share of the cross- q m and the genetic correlation is entirely due to AjqS¼0 ¼ 12 trait phenotypic correlation attributable to additive genetic cross-trait assortative mating. Substituting Eq. 6 into Eq. 2 factors. The cross-trait genetic correlation itself (qA)is yields however a function of both the cross-trait and same-trait assortative mating genetic correlations, since the corrðY ; Y Þ¼a a m 1 2 p11ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi22 12 coefficients a11 and a22 depend on the latter. þ a11a22 ð1 m11Þð1 m22ÞqA;Pleiotropy ð7Þ Within-family analysis þ c11c22qC þ e11e22qE:
The correlation between phenotypes Y1 and Y2 can The within-family correlation between traits Y1 and Y2 is i j i j thus be decomposed into parts attributable to assortative defined as corrWFðY1; Y2Þ¼corr Y1 Y1 ; Y2 Y2 ; where mating, pleiotropy, common environment, and individual individuals i and j are either non-twin siblings or DZ twins. environment. Thus,